Consider a 8-bit data word with 4 check bits that uses a Hamming single error correcting code for the 12-bit codeword. By adding another parity bit to the coded word, the Hamming code can be used to correct a single error and detect double errors. If we include this additional parity bit, then a previous 12-bit coded word becomes 001110010100P13, where Pi3 is evaluated from the exclusive-OR of the other 12 bits. This produces the 13-bit word 0011100101001 (even parity). When the 13-bit word is read from memory the check bits are evaluated and so is the parity P over the entire 13 bits. If P 0 the parity is correct (even parity), but if P = 1, then the parity over the 13 bits is incorrect (odd parity). Letting S = 83828180 be the binary representation of the syndrome resulting from evaluation of the Hamming single error correcting parity check equations, then the following four cases can occur If S = 0 and P-0: No error occurred. If S0 and P = 1: A single error occurred, which can be corrected. If S 0 and P-0: A double error occurred, which is detected but cannot be corrected. If S = 0 and P-1: Note that this scheme cannot detect more than two errors An error occurred in the 13 bit a. It is necessary to formulate the Hamming code for four data bits, dz, ds, de, d7, together (1) Assign the parity bits and show the resulting 7-bit composite codeword for the data (2) Evaluate three parity check equations for the syndrome 8281so assuming no error has (3) Assume and error in bit ds occurred during a write into memory. Show how the error (4) Add parity bit Ps to include a double-error detection capability in the code. Assume b. Given the 8-bit data word 01011011, generate the 13-bit codeword for the Hammig code c. Given the -bit data word 11001001010, generate the 16-bit codeword for the Hamming d. How many parity check bits must be included with the data word to achieve single error with three parity bits pi, p2, and pA word 0010 occurred in the bit is detected and corrected that errors occurred in bits p2 and ds. Show ho te double error is detected that corrects single errors and detects double errors code that corrects single errors and detects double errors. correction and double error detection when the data word contains: (1) 16 bits; (2) 32 bits; (3) 48 hits